Question

in the diagram below, \\( \overline { x y } \\) is parallel to \\( \overline { u v } \\). if \\( u v = 11 \\), \\( w y = 4.8 \\), and \\( v y = 4 \\), find the length of \\( \overline { x y } \\). figures are not necessarily drawn to scale.

Explanation:

Step1: Use the basic proportionality theorem (Thales' theorem)

Since \(\overline{XY}\parallel\overline{UV}\), we have \(\triangle WXY\sim\triangle WUV\). Then \(\frac{XY}{UV}=\frac{WY}{WU}\). First, find \(WU = WY+VY\). Given \(WY = 4.8\) and \(VY = 4\), so \(WU=4.8 + 4=8.8\).

Step2: Substitute values into the proportion

We know \(UV = 11\), \(WY = 4.8\), and \(WU = 8.8\). From \(\frac{XY}{UV}=\frac{WY}{WU}\), substituting gives \(\frac{XY}{11}=\frac{4.8}{8.8}\).

Step3: Solve for \(XY\)

Cross - multiply: \(8.8XY=11\times4.8\). Then \(XY=\frac{11\times4.8}{8.8}\). Simplify \(\frac{11\times4.8}{8.8}=\frac{11\times4.8}{11\times0.8}\) (since \(8.8 = 11\times0.8\)). Cancel out the \(11\) terms, and \(XY=\frac{4.8}{0.8}=6\).

Answer:

\(6\)